总变差正则化模型的区域分解算法及其医学图像应用

项目名称： 总变差正则化模型的区域分解算法及其医学图像应用

项目编号： No.11501413

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 常慧宾

作者单位： 天津师范大学

项目金额： 18万元

中文摘要： 本项目研究总变差正则化模型的区域分解算法，用于实现大规模与高复杂度的图像处理问题的并行计算，并将该算法应用于医学图像处理。总变差正则化模型的目标泛函区别于传统问题的二次泛函，具有不可微性和不可加性，为算法设计和理论分析带来巨大挑战。本项目围绕最具代表性的Rudin-Osher-Fatemi(ROF)模型开展研究。首先，研究与ROF模型等价的对偶问题，设计单水平和两水平重叠型区域分解算法，即求解凸集约束的可微凸泛函极小化问题，可以克服ROF模型目标泛函的不可微性和不可加性导致的困难；同时，研究算法的收敛性并估计算法收敛速度。其次，通过借助拉格朗日乘子建立鞍点问题来研究ROF模型原问题的非重叠型区域分解算法。最后探讨将算法应用到医学图像恢复及分割等问题。

中文关键词： 总变差正则化模型；区域分解算法；变分图像处理；医学图像

英文摘要： In this project, we plan to study the domain decomposition methods (DDMs) for the total variation regularized models in order to process the images with large scales and high complexities by parallel computing, and extend the researches to medical image processing. The objective functionals of such kind of models are commonly non-differential and non-additive, that leads to great challenges for the algorithms designing and theoretical analyses. We will mainly investigate the most representative Rudin-Osher-Fatemi (ROF) model. Firstly, we will study the dual problem of ROF model and develop the one-level and two-level DDMs to solve the deduced convex minimization problem with the convex set constraint, that will overcome the difficulties caused by the non-differentiability and non-additivity. Meanwhile, both the related convergence and an estimate of convergence rate will be derived. Secondly, we will study the non-overlapping DDMs for the primal problem, where a saddle point problem is built upon the Lagrangian multiplier. Finally, the DDMs will be extended to restorations and segmentations for medical images.

英文关键词： Total variation regularized models;Domain decomposition methods;Variational image processing;Medical images